A Resource for Free-standing Mathematics QualificationsQuadratic Graphs
Quadratic graphs have equations of the form:
where a, b, c are positive or negative constants
(b and/or c could also be zero)
To draw a quadratic graph from its equation, you need to calculate and plot points.
You need to plot enough points to give the shape of the curve.
Example gives these points:
In this case it is useful to work out an extra point:
When x = 1.5, y = 1.52 – 3 1.5 = – 2.25 This is the lowest point on the curve.
All quadratic equations have this characteristic shape.
When a ispositive, the curve has a minimum point like this one.
When a isnegative, the curve is the other way up and has a maximum point.
Example gives the following points:
To help get the shape right near the highest point, it is useful to work out extra points:
eg when x = 0.5, y = 5 + 2 0.5 – 4 0.52 = 5
when x = 0.25, y = 5 + 2 0.25 – 4 0.252 = 5.25 (the maximum value of y)
The graph is shown below.
Try these….
1a)Complete the table:
b)On the grid below draw and label the graphs of the following:
y = x2y = 2x2y = 3x2y = – x2y = – 2x2 y = – 3x2
c)Write down what you notice about your graphs.
2a)Complete the table below for
b)On the grid below plot the points from the table, but do not join them yet.
c)Find the value of y when x = 1.25 and plot this point on the grid.
d)Join the points with a smooth curve.
e)Use your graph to solve the following equations:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
3a)Draw the graph of for values of x between – 1 and 5
b)What is the maximum point on the curve?
c)Use your graph to solve the following equations:
(i) (ii)
(iii) (iv)
4a)Draw the graph of for values of x between – 2 and 5
b)Find the coordinates of the minimum point on the curve.
c)Use your graph to solve the following equations:
(i) (ii)
(iii) (iv)
(v) (vi)
5a)Draw the graph of for values of x between – 4 and 3
b)Give approximate coordinates for the minimum point on the curve.
c)Use your graph to solve the following equations:
(i) (ii)
(iii) (iv)
(v)
d)Explain how you can tell from the graph that the equation has no solutions.
6a)Draw the graph of for values of x between – 4 and 3
b)Estimate the coordinates of the maximum point on the curve.
c)Use your graph to solve the following equations:
(i) (ii)
(iii) (iv)
(v) (vi)
d)Explain how you can tell from the graph that the equation has no solutions.
UnitIntermediate Level, Using algebra, functions and graphs
Skills used in this activity:
- Drawing graphs of quadratic functions
- Using graphs to find the solutions to quadratic equations.
Notes
This activity can be used to introduce quadratic graphs or as a revision exercise at the end of the course.
The accompanying Powerpoint presentation includes the examples that are given on pages 1 and 2. The questions on pages 2 and 3 can be done on the worksheet, but those on page 5 expect students to draw the graphs on graph paper. Alternatively, students can use graphic calculators to answer all of the questions.
Answers
1a)
b)
c)Possible answers include:
All curves pass through the origin. As the coefficient of x2 increases, the curve becomes steeper.
Positivex2 terms give ashaped curve, whilst negative x2 terms give ashaped curve.
2a)
b)see graph belowc) - 6.125
d)
3a)
4a)
5a)
d) is equivalent to .
The curve does not cross the line y = - 10, so there are no solutions.
6a)
b)(- 0.5, 9.5)
c)(i) -2.7, 1.7(ii) - 1.6, 0.6(iii) - 3.8, 2.8
(iv) -1, 0(v) -2.2, 1.2(vi) - 3.6, 2.6
d) is equivalent to .
The curve does not cross the line y = 12, so there are no solutions.
The Nuffield Foundation
1